Let \(n\ge 1\)  be a number. Let \(\Gamma _n\)  be the graph on \(\mathbb {R}^n\)  connecting points of rational Euclidean distance. Komjáth [1]} proved that in ZFC, all graphs \(\Gamma _n\)  have countable chromatic number; the cases \(n=2\)  and \(n=3\)  are easier and have been known much earlier [2]} [3]}. Difficulty of the proofs greatly increases with \(n\) . The main theorem of this paper shows why this is so.
